She hates that I still talk to my ex

I am a 28-year-old guy and broke up eight months ago with someone I dated for almost three years. My current girlfriend and I have been in a relationship for around two months, and she resents the fact that my ex and I are still close friends. I spoke about this before we got together and she said she was okay with it, but it is increasingly obvious that she is not. She starts arguments with me whenever she finds out that I have been chatting with my ex and creates a lot of issues about what I discuss about my personal life. I have tried my best to explain that what I share with my ex is a pure friendship and not a romantic interest, but she doesn’t believe me. I am so tired of this that I am considering ending this relationship which is sad because I genuinely love my girlfriend. Is there any way of saving it?
What your girlfriend is feeling needs to be acknowledged and accepted, before you think of defending yourself. If she is making it clear that she can’t handle your friendship with your ex, the way to handle it is to give her time to come to terms with your life because you have only been together for a short time. Her insecurity is real and understandable, given your history with your ex. You can save this relationship by finding a way of balancing your past and 
present, and making your girlfriend feel secure with you. Talk about why she is important, acknowledge that chatting with your ex can be painful for her, and ask if there is anything you can do to minimise her feelings of abandonment in those moments. This relationship can be saved only with patience and understanding. 

How do I know if my boyfriend is being honest with me about his feelings?
You can judge his actions rather than his words and spend more time with him until you understand him better. Trust your instinct.